Sounds like a great plan, visiting the Grand Canyon in the Fall. Just remember the elevation is much higher than you might think it is.
With a limited budget and starting out, it is understandable that you wish to spend money that won't be thrown away later and works towards the future in enjoying your TC.
Your Lance will do you well down into the 20s, at night. Lower than that and you will need to follow a good regime for keeping water in your water lines.
This is a list I would recommend for you to start investing, in this order (many have been suggested by many knowledgable people earlier in this thread).
1. Insure you have a working Smoke and Propane Detectors (test before each trip)
2. Invest in a good Down Comforter (I get mine at Kohls! Plus a stocking cap and light 99 cent flannel gloves to sleep with.)
3. Charge your battery fully before leaving home.
4. Invest in a Wave or Plat Cat Heater (I have a Wave heater and love it. Best prices, for them, in the spring)
5. Purchase a Generator that will power your largest power appliance. (Be nice to friends and fellow campers and purchase a quality and low db rated generator)
6. Purchase a high amp aux Battery Charger.
7. After you learned how to manage batteries, by abusing your initial battery, invest in a high AmpH battery, minimum 2. (Either 6v or 12v)
8. Invest in a 200w Solar System with option to expand, as you discover your own power demands. (If you are planning on 4 season camping, opt for more Watts. I have 300w on the roof of my TC).
If you are planning on being in the camper for trips in cold/snowy weather. Be familiar with winterizing your water system quickly. By having an air compressor, can provide quick results. But have RV Antifreeze, by pass for H2O heater and hoses to pump into system via your existing water pump and know how to use them. Oh, yeah. Have a good filled Water Jug (Wal-Mart 6.2g Blue Jug) to use if you need to dump your on board water and use the jug for emergency.
Have fun and in time you will have built yourself a great multi season camper and have the experience to do it!
(in cold weather, be careful about introducing water/humidity into your camper, via boiling water, showers, ect. If you do, do it early in the day to allow the water molecules to escape through doors opening and closing.)
b